Lihat sumber Papirus Matematika Rhind

{{Infobox naskah
<!----------Name---------->
| form                  = Papirus
| number                = BM 10057 & 10058
| name                  = Papirus Matematika Rhind
| location              = [[British Museum]], [[London]]
<!----------Image---------->
| image                 = Rhind Mathematical Papyrus.jpg
| width                 = 300px
| caption               = Sebagian dari Papirus Rhind
<!----------General---------->
| Also known as         =
| Type                  = 
| Date                  = [[Period Menengah Kedua Mesir]]
| Place of origin       = [[Thebes]]
| Language(s)           = [[bahasa Mesir]] ([[Hieratic]])
| Scribe(s)             =
| Author(s)             =
| Compiled by           =
| Illuminated by        =
| Patron                =
| Dedicated to          =
<!----------Form and content---------->
| Material              =
| Size                  = Panjang: {{convert|536|cm|in}}<br>Lebar: {{convert|32|cm|in}}
| Format                =
| Condition             =
| Script                =
| Contents              =
| Illumination(s)       =
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| Exemplar(s)           =
| Previously kept       =
| Discovered            =
| Other                 = BM/Big no. AE 10058, Reg&nbsp;no.&nbsp;{{British-Museum-db|1865,0218.3|id=110036}}
| below                 = 
}}

'''Papirus Matematika Rhind''' ({{lang-en|'''Rhind Mathematical Papyrus'''}} disingkat '''RMP'''; juga diberi kode: '''Papyrus [[British Museum]] 10057''', dan '''pBM 10058''') adalah sebuah naskah kuno berisi catatan matematika dari budaya [[Mesir kuno]] yang ditulis di atas lembaran [[papirus]]. Dianggap sebagai contoh terbaik kemajuan [[matematika]] [[Mesir kuno]]. Dinamai menurut [[Alexander Henry Rhind]], seorang pencinta barang antik asal [[Skotlandia]], yang membeli naskah papirus itu pada tahun 1858 di [[Luxor]] ([[Thebes]]), [[Mesir]]; tampaknya diketemukan pada suatu ekskavasi ilegal di sekitar [[Ramesseum]]. Diperkirakan berasal dari tahun 1650 SM. [[British Museum]], yang menyimpan sebagian besar papirus itu, memperolehnya pada tahun 1865 bersama dengan ''[[Egyptian Mathematical Leather Roll]]'', yang juga dimiliki oleh Henry Rhind;<ref name="Clagett">Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0-87169-232-0</ref> selain itu ada pula beberapa fragmen kecil yang disimpan di [[Brooklyn Museum]] di [[New York]]<ref name="Spalinger"/><ref>{{cite web| title=Collections: Egyptian, Classical, Ancient Near Eastern Art: Fragments of Rhind Mathematical Papyrus | publisher=Brooklyn Museum | url=http://www.brooklynmuseum.org/opencollection/objects/118304/Fragments_of_Rhind_Mathematical_Papyrus | accessdate=November 1, 2012}}</ref> dan sebuah bagian tengah berukuran 18&nbsp;cm yang hilang. Salah satu dari dua Papirus Matematika kuno yang terkenal, selain [[Papirus Matematika Moskwa]]. Papirus Rhind berukuran lebih besar daripada Papirus Moskwa, meskipun usianya lebih muda dari Papirus Moskwa.<ref name="Spalinger">Anthony Spalinger, The Rhind Mathematical Papyrus as a Historical Document, Studien zur Altägyptischen Kultur, Bd. 17 (1990), pp. 295-337, Helmut Buske Verlag GmbH</ref>

Papirus Matematika Rhind bertarikh dari [[Periode Menengah Kedua Mesir]] dalam sejarah [[Mesir kuno]]. Disalin oleh seorang jurutulis bernama [[Ahmes]] (''yaitu,'' Ahmose; ''Ahmes'' merupakan ejaan yang lebih kuno yang lebih disukai oleh sejarawan matematika), dari teks yang sekarang sudah hilang yang berasal dari zaman pemerintahan [[Firaun]] [[Amenemhat III]] ([[Dinasti kedua belas Mesir]]). Ditulis dalam tulisan [[hieratic]], naskah Mesir kuno ini berukuran tingginya 33&nbsp;cm dan terdiri dari sejumlah bagian yang seluruhnya berukuran sepanjang 5 meter. Papirus itu mulai ditransliterasi dan diterjemahkan bagian matematikanya pada akhir abad ke-19. Pada tahun 2008 bagian terjemahan matematika masih belum lengkap dalam beberapa aspek. Dokumen itu memuat tarikh "Tahun ke-33 raja bangsa [[Hyksos]], [[Apepi I|Apepi I (Apophis)]] dan juga memmuat bagian di sisi baliknya (''verso'') bertanggal "Tahun ke-11'' rupanya dari raja penerusnya, [[Khamudi]].<ref>cf. Thomas Schneider's paper 'The Relative Chronology of the Middle Kingdom and the Hyksos Period (Dyns. 12-17)' in Erik Hornung, Rolf Krauss & David Warburton (editors), Ancient Egyptian Chronology (Handbook of Oriental Studies), Brill: 2006, p.194-195</ref>
<!--
In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets". He continues with: 
<blockquote>This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre (?). The scribe Ahmose writes this copy.<ref name="Clagett" />
</blockquote>

Several books and articles about the Rhind Mathematical Papyrus have been published, and  a handful of these stand out.<ref name="Spalinger"/> The Rhind Papyrus was published in 1923 by Peet and contains a discussion of the text that followed Griffith's Book I, II and III outline <ref>[[Thomas Eric Peet|Peet, Thomas Eric]]. 1923. ''The Rhind Mathematical Papyrus, British Museum 10057 and 10058''. London: The University Press of Liverpool limited and Hodder & Stoughton limited</ref> Chase published a compendium in 1927/29 which included photographs of the text.<ref>Chace, Arnold Buffum. 1927-1929. ''The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations''. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7</ref> A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute.<ref>Robins, R. Gay, and Charles C. D. Shute. 1987. ''The Rhind Mathematical Papyrus: An Ancient Egyptian Text''. London: British Museum Publications Limited. ISBN 0-7141-0944-4</ref>

==Book I==
{{Main| Rhind Mathematical Papyrus 2/n table | Egyptian fraction }}
The first part of the Rhind papyrus consists of reference tables and a collection of 20 arithmetic and 20 algebraic problems.  The problems start out with simple fractional expressions, followed by completion (''sekhem'') problems and more involved linear equations ([[Moscow_Mathematical_Papyrus#Aha_problems|''aha'' problems]]).<ref name="Spalinger"/>

The first part of the papyrus is taken up by the [[Rhind Mathematical Papyrus 2/n table|2/''n'' table]]. The  fractions 2/''n'' for odd ''n'' ranging from 3 to 101 are expressed as sums of [[Egyptian fraction|unit fractions]]. For example <math>2/15 = 1/10 + 1/30 </math>. The decomposition of 2/''n'' into unit fractions is never more than 4 terms long as in for example <math>2/101 = 1/101 + 1/202 + 1/303 + 1/606</math>.

This table is followed by a list of fraction expressions for the numbers 1 through 9 divided by 10.  For instance the division of 7 by 10 is recorded as:
: 7 divided by 10 yields 2/3 + 1/30

After these two tables, the scribe recorded 84 problems altogether and problems 1 through 40 which belong to Book I are of an algebraic nature.

Problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 and 1 + 2/3 + 1/3 by different fractions.
Problems 21–23 are problems in completion, which in modern notation is simply a subtraction problem. The problem is solved by the scribe to multiply the entire problem by a [[least common multiple]] of the denominators, solving the problem and then turning the values back into fractions. Problems 24–34 are ‘’aha’’ problems. These are [[linear equations]]. Problem 32 for instance corresponds (in modern notation) to solving  x + 1/3 x + 1/4 x = 2  for x. Problems 35–38 involve divisions of the hekat. Problems 39 and 40 compute the division of loaves and use [[arithmetic progressions]].<ref name="Clagett" />

==Book II==
[[File:Egyptian A'h-mosè or Rhind Papyrus (1065x1330).png|right|thumb|A portion of the Rhind Papyrus]]
The second part of the Rhind papyrus consists of [[geometry]] problems. Peet referred to these problems as "mensuration problems".<ref name="Spalinger"/>

===Volumes===
Problems 41 – 46 show how to find the volume of both cylindrical and rectangular based granaries. In problem 41 the scribe computes the volume of a cylindrical granary. Given the diameter (d) and the height (h), the volume V is given by:
: <math> V = [(1-1/9) d]^2 h</math>
In modern mathematical notation (and using d = 2r) this clearly equals <math> V = (8/9)^2 d^2 h = 256/81  r^2 h</math> The quotient 256/81 approximates the value of π as being ca. 3.1605.

In problem 42 the scribe uses a slightly different formula which computes the volume and expresses it in terms of the unit ''khar''.
<math> ((1+1/3)d)^2 ((2/3) h)</math> In modern mathematical notation this is equal to <math> 32/27 d^2 h = 128/27 r^2 h </math> (measured in ''khar'').
This is equivalent to <math> 256/81  r^2 h </math> measured in cubic-cubits as used in the other problem.<ref name="Clagett" />

Problem 47 gives a table with equivalent fractions for fractions of 100 quadruple hekat of grain. The quotients are expressed in terms of Horus eye fractions. The short table gives the values related to the original 100 quadruple hekat; the quantity "ro" here is a standard ancient Egyptian measure equivalent to 1/320 of a hekat.- 
: 1/10 gives 10 quadruple hekat
: 1/20 gives 5 quadruple hekat
: 1/30 gives 3 1/4 1/16 1/64 (quadruple) hekat and 1 2/3 ''ro''
: 1/40 gives 2 1/2 (quadruple) hekat
: 1/50 gives 2 (quadruple) hekat
: 1/60 gives 1 1/2 1/8 1/32 (quadruple) hekat 3 1/3 ''ro''
: 1/70 gives 1 1/4 1/8 1/32 1/64 (quadruple) hekat 2 1/14 1/21 ''ro''
: 1/80 gives 1 1/4 (quadruple) hekat
: 1/90 gives 1 1/16 1/32 1/64 (quadruple) hekat 1/2 1/18 ''ro''
: 1/100 gives 1 (quadruple) hekat<ref name="Clagett"/>

===Areas===
Problems 48 -  55 show how to compute an assortment of [[area]]s. Problem 48 is often commented on as it computes the [[Area of a disk|area of a circle]]. The scribe compares the area of a circle (approximated by an [[octagon]]) and its circumscribing square. Each side is trisected and the corner triangles are then removed. The resulting octagonal figure approximates the circle. The area of the octagonal figure is: <math>9^2 - 4 \times [\frac{1}{2}(3)(3)] = 63</math>;   Next we approximate 63 to be 64 and note that <math>64 = 8^2</math>. And we get the approximation <math>\pi (\frac{9}{2})^2 \approx 8^2</math>. Solving for π, we get the approximation <math>\pi \approx \frac{256}{81} \approx 3.1605</math> (the approximation has an error of .0189).

That this octagonal figure, whose area is easily calculated, so accurately approximates the area of the circle is just plain good luck. Obtaining a better approximation to the area using finer divisions of a square and a similar argument is not simple.<ref name="Clagett" /><ref>Don Allen <cite> Egyptian and Babylonian Mathematics from [http://www.math.tamu.edu/~dallen/history/egypt/egypt.html] </cite></ref>
Other problems show how to find the area of rectangles, triangles and trapezoids.

===Pyramids===
The final five problems are related to the slopes of [[pyramid]]s. 
A [[seked]] problem is reported by:<ref name=Maor-7/>
: If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its ''seked''?"
The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.<ref name=Maor-7>{{Cite book|title=Trigonometric Delights|first=Eli|last=Maor|year=1998|publisher=[[Princeton University Press]]|isbn=0-691-09541-8|page=20}}</ref>

==Book III==
The third part of the Rhind papyrus consists of a collection of  84 problems.<ref name="Spalinger"/> 
Problem 61 consists of 2 parts. Part 1 contains multiplications of fractions. Part b gives a general expression for computing 2/3 of 1/n, where n is odd. In modern notation the formula given is 
: <math> \frac{2}{3n} = \frac{1}{2n} + \frac{1}{6n} </math>
Problems 62- 68 are general problems of an algebraic nature.  Problems 69 - 78 are all ''pefsu'' problems in some form or another. They involve computations regarding the strength of bread and or beer.<ref name="Clagett" />

Problem RMP 79' sums five terms in a [[geometric progression]]. It is a multiple of 7 riddle, which would have been written in the Medieval era as, "[[As I was going to St Ives|Going to St. Ives]]" problem.<ref name="Spalinger"/>
Problems 80 and 81 compute [[Horus eye]] fractions of henu (or hekats). Problem 81 is followed by a table. The last three problems 82 - 84 compute the amount of feed necessary for fowl and oxen.<ref name="Clagett" />
-->
== Lihat pula ==
* [[Lahun Mathematical Papyri]]
* [[Akhmim wooden tablet]]
* [[Berlin Papyrus 6619]]
* [[Rhind Mathematical Papyrus 2/n table]]

== Referensi ==
{{reflist}}

== Pustaka tambahan ==
* Gillings, Richard J. "Mathematics in the Time of the Pharaohs", 1972, MIT Press, Dover reprint ISBN 0-486-24315-X

== Pranala luar ==
* Allen, Don. April 2001. [http://www.math.tamu.edu/~don.allen/history/egypt/node3.html ''The Ahmes Papyrus''] and [http://www.math.tamu.edu/~don.allen/history/egypt/node5.html ''Summary of Egyptian Mathematics''].
* {{dmoz|Society/History/By_Time_Period/Ancient/Africa/Egypt/Texts|Egypt/Texts}}
* [http://www.britishmuseum.org/explore/highlights/highlight_objects/aes/r/rhind_mathematical_papyrus.aspx British Museum webpage on the Papyrus.]
* O'Connor and Robertson, 2000. [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html ''Mathematics in Egyptian Papyri''].
* Truman State University, Math and Computer Science Division. '''Mathematics and the Liberal Arts:''' [http://math.truman.edu/~thammond/history/RhindPapyrus.html ''The Rhind/Ahmes Papyrus''].
* {{cite web 
  | authorlink = Eric W. Weisstein
  | title = Rhind Papyrus
  | publisher = MathWorld–A Wolfram Web Resource
  | url = http://mathworld.wolfram.com/RhindPapyrus.html}}
* Williams, Scott W. [http://www.math.buffalo.edu/mad/index.html ''Mathematicians of the African Diaspora''], containing a page on [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html ''Egyptian Mathematics Papyri''].
* [http://www.bbc.co.uk/ahistoryoftheworld/objects/y1T3knf-T66RwWyEt_cZBw BBC audio file] ''[[A History of the World in 100 Objects]]''. (15 mins)

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